Involutions on the affine Grassmannian and moduli spaces of principal bundles

Anthony Henderson


Let \(G\) be a connected reductive group over \(\mathbb{C}\). We show that a certain involution of an open subset of the affine Grassmannian of \(G\), defined previously by Achar and the author, corresponds to the action of the nontrivial Weyl group element of \(\mathrm{SL}(2)\) on the framed moduli space of \(\mathbb{G}_m\)-equivariant principal \(G\)-bundles on \(\mathbb{P}^2\). As a result, the fixed-point set of the involution can be partitioned into strata indexed by conjugacy classes of homomorphisms \(N\to G\) where \(N\) is the normalizer of \(\mathbb{G}_m\) in \(\mathrm{SL}(2)\). In the case where \(G=\mathrm{GL}(r)\), the strata are Nakajima quiver varieties \(\mathfrak{M}_0^{\mathrm{reg}}(\mathbf{v},\mathbf{w})\) of type \(D\).

Keywords: Affine Grassmannian; moduli space; quiver variety.

AMS Subject Classification: Primary 14J60; Secondary 14M15, 17B08.

This paper is available as a pdf (656kB) file.

Thursday, December 17, 2015